Problem 34 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$654.321-123.456$$

Step-by-Step Solution

Answer

The difference is 530.865.

Step 1: Align Decimal Points

Write the numbers vertically and align them by the decimal point, so that each place value lines up correctly. This helps to ensure accuracy when subtracting each digit in the correct column.

Step 2: Subtract the Thousandths Place

Start subtracting from the rightmost digit (thousandths place). Subtract 6 from 1. Since 1 is smaller than 6, we need to borrow from the tenths place. This makes the thousandths digit 11, and we subtract 6 to get 5.

Step 3: Subtract the Hundredths Place

Move to the hundredths place. After borrowing, subtract the hundredths digit of 5 from 2. Since 2 is smaller than 5, we need to borrow from the ones place. This makes the hundredths digit 12, and we subtract 5 to get 7.

Step 4: Subtract the Tenths Place

Subtract the tenths place. After borrowing, subtract the tenths digit of 4 from 1 (since we borrowed 1 from this column, it is now reduced by 1). We need to borrow once again from the next column, which makes it 11, then subtract 4 to get 7.

Step 5: Subtract the Units Place

Subtract the ones place. We borrowed from the units place earlier, so subtract 3 from 4 (after borrowing it reads as 14), which results in 1.

Step 6: Subtract the Tens Place

Subtract the tens place. With the prior borrowing adjustment, subtract 2 from 5. This results in 3.

Step 7: Subtract the Hundreds Place

Finally, subtract the hundreds place. Subtract 1 from 6 (no borrowing is needed), resulting in 5.

Key Concepts

Decimal Subtraction

Subtracting numbers that include decimals requires careful alignment by their decimal points. When you line up the numbers vertically, each digit falls into its respective place value. This alignment ensures that you subtract corresponding digits correctly. Consider decimal subtraction like using a calculator, where the positions are crucial for accuracy. The subtraction of decimals is almost the same as whole numbers, focusing on each column separately. For example, in the problem \(654.321 - 123.456\), align the decimal points so each digit like hundreds, tens, ones, etc., are in the correct positions. This way, when you start subtracting from the rightmost to leftmost, each digit correctly corresponds to its partner in subtraction.

Place Value

Place value is essential to understand, as it represents the position of a digit in a number. Each digit's position dictates its value, ranging from units, tenths, hundredths, thousandths, and so forth, when dealing with decimals. For instance, in the problem \(654.321 - 123.456\), 6 in the number 654.321 is in the hundreds place, which gives it a value of 600. Similarly, 5 is in the tens place, valued at 50. Subtracting correctly involves understanding and working with each of these values by their place. By grasping this concept, you can ensure precision in calculating differences as you work step by step through each place value column from right to left, making sure to align and correspond each digit correctly.

Borrowing in Subtraction

Borrowing is a necessary part of subtraction when the larger digit is taken from a smaller digit. In our exercise, when subtracting numbers such as \(654.321\) and \(123.456\), you can't directly subtract some lower digits without assistance, which involves borrowing from the higher place value. For example, if you're subtracting 6 from 1, you borrow 1 from the next left digit, turning the 1 into 11. Then, you subtract the 6 from 11 to get 5.

• Align the numbers properly.
• Start from the rightmost digit.
• Borrow from the next left if needed.

Borrowing doesn't change the overall value but redistributes it across the number to help simplify calculations.

Multi-Digit Subtraction

Multi-digit subtraction involves subtracting numbers with many digits, often requiring borrowing and careful place value alignment. In our given exercise, the operation is represented as \(654.321 - 123.456\) involving subtraction across the decimal down to thousandths place. The key here is to maintain order:

• Align the numbers by decimal points.
• Move column by column, from right to left.
• Borrow where necessary to ensure a larger digit is never subtracted from a smaller digit.
• Ensure each digit corresponds to its own place value, keeping accurate subtraction results.

Multi-digit subtraction requires patience and accuracy in order to achieve the intended correct result without errors due to misalignment or incorrect borrowing adjustments.